This is very emotional for you and I am sorry you are struggling at the moment, but you also need to see it from your employer's perspective: as a Head of department and team leader, a big part of your job is to lead, coach, and support your team. That really can't be done in 2 days per week, it would have too big a negative impact on the team (and if you don't think that's the case, then maybe a leadership position is not the best role for you - which is ok too).
Asking to go down from 5 to 2 days per week is not a basis for negotiation, it's too disruptive for a business. My advice would be: ask to go down to 4 days for a specific period of time (3 months for instance), making the terms very clear on both sides so they know what to expect from you ("might not be permanent" is too unpredictable).
Also, if you keep telling them that "your 5-day job can really be done in 2 days"... at some point they will take your word for it, and quickly replace you with another person who will work more productively, essentially delivering twice as much output in the same time.
As sad as it sounds, the responsibility of your employer is to run their business effectively and productively, it is not to accommodate employees if their requests are too disruptive.
Wishing you all the best.